Algebra

The quadratic formula expresses the solution of the degree two equation ax^2 + bx +c=0 in terms of its coefficients a, b, c, where a is not equal to 0.

Algebra (from Arabical-jebr meaning "reunion of broken parts"[1]) is one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form algebra is the study of symbols and the rules for manipulating symbols[2] and is a unifying thread of all of mathematics.[3] As such, it includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. The more basic parts of algebra are called elementary algebra, the more abstract parts are called abstract algebra or modern algebra. Elementary algebra is essential for any study of mathematics, science, or engineering, as well as such applications as medicine and economics. Abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians. Much early work in algebra, as the Arabic origin of its name suggests, was done in the Near East, by such mathematicians as Omar Khayyam (1048-1131).[4][5]

Elementary algebra differs from arithmetic in the use of abstractions, such as using letters to stand for numbers that are either unknown or allowed to take on many values.[6] For example, in x + 2 = 5 the letter x is unknown, but the law of inverses can be used to discover its value: x=3. In E=mc^2, the letters E and m are variables, and the letter c is a constant. Algebra gives methods for solving equations and expressing formulas that are much easier (for those who know how to use them) than the older method of writing everything out in words.

The word algebra is also used in certain specialized ways. A special kind of mathematical object in abstract algebra is called an "algebra", and the word is used, for example, in the phrases linear algebra and algebraic topology (see below).

How to distinguish between different meanings of "algebra"

For historical reasons, the word "algebra" has several related meanings in mathematics, as a single word or with qualifiers. Such a situation, where a single word has many meanings in the same area of mathematics, may be confusing. However the distinction is easier if one recalls that the name of a scientific area is usually singular and without an article and the name of a specific structure requires an article or the plural. Thus we have:

As a single word without article, "algebra" names a broad part of mathematics (see below).

Algebra as a branch of mathematics

Algebra began with computations similar to those of arithmetic, with letters standing for numbers.[6] This allowed proofs of properties that are true no matter which numbers are involved. For example, in the quadratic equation

ax^2+bx+c=0,

a, b, c can be any numbers whatsoever (except that a cannot be 0), and the quadratic formula can be used to quickly and easily find the value of the unknown quantity x.

Before the 16th century, mathematics was divided into only two subfields, arithmetic and geometry. Even though some methods, which had been developed much earlier, may be considered nowadays as algebra, the emergence of algebra and, soon thereafter, of infinitesimal calculus as subfields of mathematics only dates from 16th or 17th century. From the second half of 19th century on, many new fields of mathematics appeared, most of which made use of both arithmetic and geometry, and almost all of which used algebra.

History

The start of algebra as an area of mathematics may be dated to the end of 16th century, with François Viète's work. Until the 19th century, algebra consisted essentially of the theory of equations. In the following, "Prehistory of algebra" is about the results of the theory of equations that precede the emergence of algebra as an area of mathematics.

The Hellenistic mathematicians Hero of Alexandria and Diophantus[13] as well as Indian mathematicians such as Brahmagupta continued the traditions of Egypt and Babylon, though Diophantus' Arithmetica and Brahmagupta's Brahmasphutasiddhanta are on a higher level.[14] For example, the first complete arithmetic solution (including zero and negative solutions) to quadratic equations was described by Brahmagupta in his book Brahmasphutasiddhanta. Later, Arabic and Muslim mathematicians developed algebraic methods to a much higher degree of sophistication. Although Diophantus and the Babylonians used mostly special ad hoc methods to solve equations, Al-Khwarizmi contribution was fundamental. He solved linear and quadratic equations without algebraic symbolism, negative numbers or zero, thus he has to distinguish several types of equations.[15]

In the context where algebra is identified with the theory of equations, the Greek mathematician Diophantus has traditionally been known as the "father of algebra" but in more recent times there is much debate over whether al-Khwarizmi, who founded the discipline of al-jabr, deserves that title instead.[16] Those who support Diophantus point to the fact that the algebra found in Al-Jabr is slightly more elementary than the algebra found in Arithmetica and that Arithmetica is syncopated while Al-Jabr is fully rhetorical.[17] Those who support Al-Khwarizmi point to the fact that he introduced the methods of "reduction" and "balancing" (the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation) which the term al-jabr originally referred to,[18] and that he gave an exhaustive explanation of solving quadratic equations,[19] supported by geometric proofs, while treating algebra as an independent discipline in its own right.[20] His algebra was also no longer concerned "with a series of problems to be resolved, but an exposition which starts with primitive terms in which the combinations must give all possible prototypes for equations, which henceforward explicitly constitute the true object of study". He also studied an equation for its own sake and "in a generic manner, insofar as it does not simply emerge in the course of solving a problem, but is specifically called on to define an infinite class of problems".[21]

The Persian mathematician Omar Khayyam is credited with identifying the foundations of algebraic geometry and found the general geometric solution of the cubic equation. Another Persian mathematician, Sharaf al-Dīn al-Tūsī, found algebraic and numerical solutions to various cases of cubic equations.[22] He also developed the concept of a function.[23] The Indian mathematicians Mahavira and Bhaskara II, the Persian mathematician Al-Karaji,[24] and the Chinese mathematician Zhu Shijie, solved various cases of cubic, quartic, quintic and higher-order polynomial equations using numerical methods. In the 13th century, the solution of a cubic equation by Fibonacci is representative of the beginning of a revival in European algebra. As the Islamic world was declining, the European world was ascending. And it is here that algebra was further developed.

Areas of mathematics with the word algebra in their name

Some areas of mathematics that fall under the classification abstract algebra have the word algebra in their name; linear algebra is one example. Others do not: group theory, ring theory, and field theory are examples. In this section, we list some areas of mathematics with the word "algebra" in the name.

Elementary algebra, the part of algebra that is usually taught in elementary courses of mathematics.

Elementary algebra

Elementary algebra is the most basic form of algebra. It is taught to students who are presumed to have no knowledge of mathematics beyond the basic principles of arithmetic. In arithmetic, only numbers and their arithmetical operations (such as +, −, ×, ÷) occur. In algebra, numbers are often represented by symbols called variables (such as a, n, x, y or z). This is useful because:

It allows the general formulation of arithmetical laws (such as a + b = b + a for all a and b), and thus is the first step to a systematic exploration of the properties of the real number system.

It allows the reference to "unknown" numbers, the formulation of equations and the study of how to solve these. (For instance, "Find a number x such that 3x + 1 = 10" or going a bit further "Find a number x such that ax + b = c". This step leads to the conclusion that it is not the nature of the specific numbers that allows us to solve it, but that of the operations involved.)

It allows the formulation of functional relationships. (For instance, "If you sell x tickets, then your profit will be 3x − 10 dollars, or f(x) = 3x − 10, where f is the function, and x is the number to which the function is applied".)

Polynomials

A polynomial is an expression that is the sum of a finite number of non-zero terms, each term consisting of the product of a constant and a finite number of variables raised to whole number powers. For example, x2 + 2x − 3 is a polynomial in the single variable x. A polynomial expression is an expression that may be rewritten as a polynomial, by using commutativity, associativity and distributivity of addition and multiplication. For example, (x − 1)(x + 3) is a polynomial expression, that, properly speaking, is not a polynomial. A polynomial function is a function that is defined by a polynomial, or, equivalently, by a polynomial expression. The two preceding examples define the same polynomial function.

Two important and related problems in algebra are the factorization of polynomials, that is, expressing a given polynomial as a product of other polynomials that can not be factored any further, and the computation of polynomial greatest common divisors. The example polynomial above can be factored as (x − 1)(x + 3). A related class of problems is finding algebraic expressions for the roots of a polynomial in a single variable.

Teaching algebra

It has been suggested that elementary algebra should be taught as young as eleven years old,[27] though in recent years it is more common for public lessons to begin at the eighth grade level (≈ 13 y.o. ±) in the United States.[28]

Since 1997, Virginia Tech and some other universities have begun using a personalized model of teaching algebra that combines instant feedback from specialized computer software with one-on-one and small group tutoring, which has reduced costs and increased student achievement.[29]

Abstract algebra

Abstract algebra extends the familiar concepts found in elementary algebra and arithmetic of numbers to more general concepts. Here are listed fundamental concepts in abstract algebra.

Sets: Rather than just considering the different types of numbers, abstract algebra deals with the more general concept of sets: a collection of all objects (called elements) selected by property specific for the set. All collections of the familiar types of numbers are sets. Other examples of sets include the set of all two-by-two matrices, the set of all second-degree polynomials (ax2 + bx + c), the set of all two dimensional vectors in the plane, and the various finite groups such as the cyclic groups, which are the groups of integers modulon. Set theory is a branch of logic and not technically a branch of algebra.

Binary operations: The notion of addition (+) is abstracted to give a binary operation, ∗ say. The notion of binary operation is meaningless without the set on which the operation is defined. For two elements a and b in a set S, a ∗ b is another element in the set; this condition is called closure. Addition (+), subtraction (-), multiplication (×), and division (÷) can be binary operations when defined on different sets, as are addition and multiplication of matrices, vectors, and polynomials.

Identity elements: The numbers zero and one are abstracted to give the notion of an identity element for an operation. Zero is the identity element for addition and one is the identity element for multiplication. For a general binary operator ∗ the identity element e must satisfy a ∗ e = a and e ∗ a = a. This holds for addition as a + 0 = a and 0 + a = a and multiplication a × 1 = a and 1 × a = a. Not all sets and operator combinations have an identity element; for example, the set of positive natural numbers (1, 2, 3, ...) has no identity element for addition.

Inverse elements: The negative numbers give rise to the concept of inverse elements. For addition, the inverse of a is written −a, and for multiplication the inverse is written a−1. A general two-sided inverse element a−1 satisfies the property that a ∗ a−1 = 1 and a−1 ∗ a = 1 .

Associativity: Addition of integers has a property called associativity. That is, the grouping of the numbers to be added does not affect the sum. For example: (2 + 3) + 4 = 2 + (3 + 4). In general, this becomes (a ∗ b) ∗ c = a ∗ (b ∗ c). This property is shared by most binary operations, but not subtraction or division or octonion multiplication.

Commutativity: Addition and multiplication of real numbers are both commutative. That is, the order of the numbers does not affect the result. For example: 2 + 3 = 3 + 2. In general, this becomes a ∗ b = b ∗ a. This property does not hold for all binary operations. For example, matrix multiplication and quaternion multiplication are both non-commutative.

Groups

Combining the above concepts gives one of the most important structures in mathematics: a group. A group is a combination of a set S and a single binary operation ∗, defined in any way you choose, but with the following properties:

An identity element e exists, such that for every member a of S, e ∗ a and a ∗ e are both identical to a.

Every element has an inverse: for every member a of S, there exists a member a−1 such that a ∗ a−1 and a−1 ∗ a are both identical to the identity element.

The operation is associative: if a, b and c are members of S, then (a ∗ b) ∗ c is identical to a ∗ (b ∗ c).

If a group is also commutative—that is, for any two members a and b of S, a ∗ b is identical to b ∗ a—then the group is said to be abelian.

For example, the set of integers under the operation of addition is a group. In this group, the identity element is 0 and the inverse of any element a is its negation, −a. The associativity requirement is met, because for any integers a, b and c, (a + b) + c = a + (b + c)

The nonzero rational numbers form a group under multiplication. Here, the identity element is 1, since 1 × a = a × 1 = a for any rational number a. The inverse of a is 1/a, since a × 1/a = 1.

The integers under the multiplication operation, however, do not form a group. This is because, in general, the multiplicative inverse of an integer is not an integer. For example, 4 is an integer, but its multiplicative inverse is ¼, which is not an integer.

Semigroups, quasigroups, and monoids are structures similar to groups, but more general. They comprise a set and a closed binary operation, but do not necessarily satisfy the other conditions. A semigroup has an associative binary operation, but might not have an identity element. A monoid is a semigroup which does have an identity but might not have an inverse for every element. A quasigroup satisfies a requirement that any element can be turned into any other by either a unique left-multiplication or right-multiplication; however the binary operation might not be associative.

Rings and fields

Groups just have one binary operation. To fully explain the behaviour of the different types of numbers, structures with two operators need to be studied. The most important of these are rings, and fields.

A ring has two binary operations (+) and (×), with × distributive over +. Under the first operator (+) it forms an abelian group. Under the second operator (×) it is associative, but it does not need to have identity, or inverse, so division is not required. The additive (+) identity element is written as 0 and the additive inverse of a is written as −a.

Distributivity generalises the distributive law for numbers. For the integers (a + b) × c = a × c + b × c and c × (a + b) = c × a + c × b, and × is said to be distributive over +.

The integers are an example of a ring. The integers have additional properties which make it an integral domain.

A field is a ring with the additional property that all the elements excluding 0 form an abelian group under ×. The multiplicative (×) identity is written as 1 and the multiplicative inverse of a is written as a−1.

The rational numbers, the real numbers and the complex numbers are all examples of fields.

^ abc(Boyer 1991, "Europe in the Middle Ages" p. 258) "In the arithmetical theorems in Euclid's Elements VII-IX, numbers had been represented by line segments to which letters had been attached, and the geometric proofs in al-Khwarizmi's Algebra made use of lettered diagrams; but all coefficients in the equations used in the Algebra are specific numbers, whether represented by numerals or written out in words. The idea of generality is implied in al-Khwarizmi's exposition, but he had no scheme for expressing algebraically the general propositions that are so readily available in geometry."

^(Boyer 1991, "The Arabic Hegemony" p. 229) "It is not certain just what the terms al-jabr and muqabalah mean, but the usual interpretation is similar to that implied in the translation above. The word al-jabr presumably meant something like "restoration" or "completion" and seems to refer to the transposition of subtracted terms to the other side of an equation; the word muqabalah is said to refer to "reduction" or "balancing" – that is, the cancellation of like terms on opposite sides of the equation."

^(Boyer 1991, "The Arabic Hegemony" p. 230) "The six cases of equations given above exhaust all possibilities for linear and quadratic equations having positive root. So systematic and exhaustive was al-Khwarizmi's exposition that his readers must have had little difficulty in mastering the solutions."

^Gandz and Saloman (1936), The sources of al-Khwarizmi's algebra, Osiris i, p. 263–277: "In a sense, Khwarizmi is more entitled to be called "the father of algebra" than Diophantus because Khwarizmi is the first to teach algebra in an elementary form and for its own sake, Diophantus is primarily concerned with the theory of numbers".

^Rashed, R.; Armstrong, Angela (1994). The Development of Arabic Mathematics.

^(Boyer 1991, "The Arabic Hegemony" p. 239) "Abu'l Wefa was a capable algebraist as well as a trigonometer. ... His successor al-Karkhi evidently used this translation to become an Arabic disciple of Diophantus – but without Diophantine analysis! ... In particular, to al-Karkhi is attributed the first numerical solution of equations of the form ax2n + bxn = c (only equations with positive roots were considered),"

^"The Origins of Abstract Algebra". University of Hawaii Mathematics Department.

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